The general term of a geometric series with ratio $$r\ne 1$$ is: $$$ S_n= \sum_{k=1}^n u_k = u_1\frac{r^n-1}{r-1}$$$ For $$r=1$$, the series is $$S_n = nu_1$$, because the sequence is constant.

For $$\vert r\vert\lt 1$$, the series converges and is $$$ \sum_{k=1}^{\infty} u_k = \frac{u_1}{1-r}. $$$

The form of a geometric sequence / series can be proved by induction. For the series, it can be seen as well from the identity $$$ (1+r+\dots+r^{n-1}) (r-1)= r - 1 + r^2 - r +\dots r^{n} - r^{n-1} = r^n-1$$$ after dividing it by $$(r-1)$$ when $$r\neq 1$$.

The early growth of a bacterial colony is a geometric sequence. If the bacteria do not die, the total number is a geometric series.